The function 𝑦 = 𝑓(𝑥) = 𝑥 2 − 6 is plotted along with its inverse function 𝑦 = 𝑓−1(𝑥), where
essentially the x & y axis are interchanged. These two curves will intersect at:
A. (0, 0)
B. (2, 3)
C. (2, 2) & (3, 3)
D. (−2, −2) & (3, 3)

 

 

Understanding the Function and Its Inverse

Let’s explore how to determine where a function and its inverse intersect. We’ll use the quadratic function:

f(x) = x² − 6

Its inverse, in theory, would swap the x and y variables. But since quadratic functions are not one-to-one by default, their inverses are not true functions unless we restrict the domain.

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Step-by-Step Approach

Step 1: Set the Function Equal to Its Inverse

For a function and its inverse to intersect, they must share the same x and y values, meaning:

f(x) = f⁻¹(x) ⇒ x = y

So, we solve:

y = x = x² − 6

Step 2: Rearrange into Standard Quadratic Form

x² − x − 6 = 0

Use the quadratic formula:

x = [1 ± √(1² + 4×6)] / 2 = [1 ± √25] / 2 = [1 ± 5] / 2

So the solutions are:

x = (1 + 5)/2 = 3,
   = (1 − 5)/2 = -2

Step 3: Find the Corresponding Points

Substitute x back into the function:

f(3) = 3² − 6 = 9 − 6 = 3
f(−2) = (−2)² − 6 = 4 − 6 = −2

So, the points of intersection are:

(3, 3) and (−2, −2)

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Final Answer: D. (−2, −2) & (3, 3)

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Key Takeaways

• The function and its inverse intersect where f(x) = x.
• Always restrict the domain of non-injective functions like quadratics when dealing with inverses.
• The inverse reflects the function across the line y = x.